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A337367 Sum of square end-to-end distance over all self-avoiding n-step walks on a square lattice where no adjacent points are allowed, except those for consecutive steps.

%I A337367 #23 Dec 23 2024 14:53:46
%S A337367 0,4,32,156,608,2116,6816,20844,61376,175628,491248,1349172,3650144,
%T A337367 9751532,25774672,67501556,175375136,452454276,1160098576,2958123556,
%U A337367 7505767840,18959922796,47701159264,119570463980,298719578688,743984084700,1847709517360,4576818079076,11309417827072
%N A337367 Sum of square end-to-end distance over all self-avoiding n-step walks on a square lattice where no adjacent points are allowed, except those for consecutive steps.
%C A337367 The corresponding number of n-step walks is given in A173380.
%D A337367 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes the sequence A173380).
%H A337367 Sequence Fans Mailing list, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2010-November/006470.html">discussion of the sequence A173380</a>, November 2010.
%e A337367 The allowed 4-step walks with their associated end-to-end square distances are:
%e A337367 .
%e A337367          + 10
%e A337367 4        |        8              8      8           16
%e A337367 +--+     +     +--+              +      +    X--+---+---+---+
%e A337367    |     |     |          10     |      |
%e A337367    +     +     +     +--+--+  +--+      +        +--+ 10      + 10
%e A337367    |     |     |     |        |         |        |            |
%e A337367 X--+  X--+  X--+  X--+     X--+   X--+--+  X--+--+   X--+--+--+
%e A337367 .
%e A337367 The eight non-straight walks sum to 68, and these can be walked in eight ways on the square lattice. The remaining straight walk can be walking in four ways. Thus a(4) = 68 * 8 + 16 * 4 = 608.
%Y A337367 Cf. A173380, A001411.
%K A337367 nonn,walk
%O A337367 0,2
%A A337367 _Scott R. Shannon_, Aug 25 2020